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Problem. Find L[*J*_{0}(t)] where *J*_{0}(t) is the Bessel function of order zero.

Solution.

Method 1, using series. *J*_{0}(t) is given by

Applying the formula

to each of the terms we get

Now it happens that the quantity in brackets is the binomial expansion of

Thus

Method 2, using differential equations. The function *J*_{0}(t) satisfies Bessel’s differential
equation which reduces in the case of *J*_{0}(t) to

We shall now take the Laplace transform of both sides of 1). Letting y = L [*J*_{0}(t)] and using the
theorems

together with *J*_{0}(0) = 1, *J*_{0}'(0) = 0, we obtain

which gives

Expanding and rearranging we obtain

or

We now integrate both sides to obtain

or

Thus

where c is an arbitrary constant. We now determine the value of the constant c as follows: Multiply 7) by s and take the limit as s to give

Now since

we deduce by the initial-value theorem that c = 1. Thus we obtain

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